Bayesian approaches for image restoration

BAYESIAN APPROACHES FOR IMAGE RESTORATION WEIMIAO YU (M. ENG., NUS AND XJTU) A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE SEPT. 2007 To my parents ACKNOWLEDGMENTS I wish to express my gratitude and appreciation to my supervisor, A/Prof. Kah Bin LIM for his instructive guidance and constant personal encouragement during every stage of my PhD study. I gratefully acknowledge the financial support provided by the National University of Singapore through Research Scholarship that makes it possible for me to finish this study. My gratitude also goes to Mr. Yee Choon Seng, Mrs. Ooi, Ms. Tshin and Mr. Zhang for their helps on facility support in the laboratory so that my research could be completed smoothly. It is also a true pleasure for me to meet many nice and wise colleagues in the Control and Mechtronics Laboratory, who made the past four years exciting and the experience worthwhile. I am sincerely grateful for the friendship and companionship from Chu Wei, Duan Kaibo, Feng Kai, Zhang Han, Liu Zheng, Long Bo, Hu Jiayi and Zhou Wei, etc. Finally, I would like to thank my parents, brother and sister for their constant love and endless support through my student life. My gratefulness and appreciation cannot be expressed in words. Weimiao YU Bayesian Approaches for Image Restoration TABLE OF CONTENTS Acknowledgments I Table of Contents . II Summary .V List of figures . VII List of tables . XII List of symbols XIV list of abbreviation XVI Chapter Introduction and Literature Review 1.1. Introduction 1.2. Objectives 1.3. Literature Review 1.3.1. Image Identification 1.3.2. Image Restoration .11 1.4. Contibutions and Organization of the Dissertation .17 Chapter Mathematical Model for Image, Noise and Blurring Process . 20 2.1. Image and Image Representation .20 2.2. General Linear Model and Kernel Trick 24 2.3. Definition of Discontinuity .26 2.4. Markov Random Field and Bayesian Inference 30 2.5. Fidelity Criteria of Image Quality .36 Chapter Bayesian Inference of Edge Identification in 1-D Piecewise Constant Signal 39 3.1. Markov 1-D Piecewise Signal with Step Edge 39 3.2. Step Edge Detection Based on Posterior Evidence in 1-D .42 3.2.1. Estimation of the Edge Points with Known Edge Number 42 3.2.2. Posterior Evidence Based on Sequent Model Select .46 3.3. Experimental Results .51 3.3.1 Signal Generation and prior Distribution of the Models Orders .51 3.3.2 Experiments for the Single Edge Point Model .54 ME Department of National University of Singapore II Weimiao YU Bayesian Approaches for Image Restoration 3.3.3. Experiments for the Multi Edge Point Model .59 3.3.4. Application of Edge Detection for 2-D Image 62 3.4. Summary 64 Chapter Sparse Probabilistic Linear Model and Relevance Vector Machine for Piecewise Linear 1-D Signal 65 4.1. Relevance Vector Machine 65 4.1.1. Likelihood Function and priori Probability 67 4.1.2. Posterior Distribution and Evidence 69 4.2. Occam’s Razor and Automatic Relevance Determination .72 4.3. Experiments of Sparse Bayesian Kernel for 1-D Piecewise Linear Signals .74 4.4 Summary .81 Chapter Sparse Kernel Image Noise Removal with Edge Preservation 83 5.1. Relevance Vector Machine in Image Restoration .83 5.1.1. Window and Local Piecewise Linear Assumptions .83 5.1.2. Inverse of the Hessian Matrix 86 5.2. Local Regularization and Global Cost Function .87 5.2.1. Selecting Kernel Function 87 5.2.2. Window Size and Kernel Matrix .88 5.2.3. Hyper-parameter Tuning 89 5.4. Experimental Results .97 5.5. Summary . 111 Chapter Statistical Approach for Motion Blur Identification . 113 6.1. Introduction . 113 6.2. Analysis of the Blur Effect in the Derivative Image . 114 6.2.1. The Derivative of Blurred Image . 114 6.2.2. The Edge and Smooth Regions in the Derivative Image 116 6.3. Blur Identification in Spatial Domain . 119 6.3.1. Extraction of the PSF Extent . 119 6.3.2. Parameter Identification 121 6.4. Experimental Results 123 6.4.1. Experimental Images and PSFs 123 6.4.2. Identification of the PSF Extent 125 ME Department of National University of Singapore III Weimiao YU Bayesian Approaches for Image Restoration 6.4.3. Parameter Extraction of the Experimental Images 129 6.4. Summary . 133 Chapter Compact Discrete Polar Transform for Rotational Blurred Images 135 7.1. Introduction . 135 7.2. Separation of the Spatially Variance and Invariance 136 7.2. Discrete Coordinates Transform between Cartesian and Polar . 138 7.3. Pixel Mapping between the Two Coordinates and the Interpolation of the Virtual Points . 144 7.4. Optimization of the PRR 146 7.5. Simulation and Experiments 148 7.6. Summary . 154 Chapter Conclusion . 156 Bibliography . 159 ME Department of National University of Singapore IV Weimiao YU Bayesian Approaches for Image Restoration SUMMARY Recovering signals (1-D/2-D) from their degraded and noisy version is generally called digital signal/image restoration or reconstruction. Some of the signals themselves contain discontinuities, which encode important, crucial and significant information. The removal of the noise and the preservation of the discontinuities are conflicting interests. The Bayesian approaches for discontinuity identification and edge preserved noise removal in 1-D/2-D digital signals are discussed and studied in this dissertation. Bayesian model selection is robust, however expensive in calculation due to the “curse of dimensionalities”. An approach is proposed in this work to reduce the calculation burden in Bayesian model selection. By increasing the model order sequentially according to a confidence level, the model order may be determined and evaluating the posterior distribution directly can be avoided. Some simplifications of the calculation for the normalization constant make it more efficient. Since the model priori distribution is not uniform, the so-called posterior evidence is suggested for the model selection. The simulations proved its robustness and accuracy. Bayesian Kernel approach based on sparse kernel learning is presented for both 1-D and 2-D discontinuous signal denoising. The image is assumed to be locally piecewise linear. The constructed kernel matrix is a hybrid of linear and quadratic in 2-D lattice. The cost function and hyper parameter tuning are studied in detail. The presented cost function is proven to only have one global minimum; therefore the problem of ill-conditioning is successfully solved. Experimental results show that the proposed method is robust and can preserve the discontinuities in its formulation. The proposed approach also achieved an excellent restoration results compared with other recent methods, some of the errors are much smaller than existing methods when Signal to Noise Ratio (SNR) is low. The Bayesian approach for blur identification in the spatial domain is proposed. This approach can identify extent of the Point Spread Function (PSF) and its parameter. No assumption is made on the ME Department of National University of Singapore V Weimiao YU Bayesian Approaches for Image Restoration shape of the PSF, so it can be applied for any shape of PSF. The image is decomposed into smooth region and edge region. Four linear motion blur PSFs are applied to blur the experimental images, thus they are successfully identified by the proposed approach. The assumptions are validated by the experimental results. The results also show it is robust and promising in blur identification. Finally, the discrete formulation of compact coordinate transformation is presented. Under-sampling phenomena are studied and discussed. The constraints of coordinates transform are relaxed in the discrete formulation. The optimization of the compactness and a cost function is proposed. The experiments and simulations show that the presented method can solve the spatially variant problem of rotational blurred image restoration. ME Department of National University of Singapore VI Weimiao YU Bayesian Approaches for Image Restoration LIST OF FIGURES Figure 1.1 The Procedure of Image Restoration . Figure 2.1 The Coordinates System and the Support of the Image . 23 Figure 2.2 Quadratic and Linear Kernel Function 25 Figure 2.3 Different Discontinuities in the 1-D Piecewise Linear Function 29 Figure 2.4 Quadratic Cost Function and δ Cost Function 35 Figure 3.1 Piecewise Constant Signal with Different Values of ρ (No noise) 52 Figure 3.2 Piecewise Constant Signal with Different Noise Level . 52 Figure 3.3 priori Distribution of the Model Order Given ρ = 0.02 . 53 Figure 3.4 priori Distribution of the Model Order for Given ρ = 0.1 . 54 Figure 3.5 Signal with Single Edge Point and the Confidence of Edge Location (Edge=102 and σ=20) 55 Figure 3.6 Confidence of Edge Location at Different Noise Level (Edge=130) . 56 Figure 3.7 Accuracy and the Magnitude of the Edges under Difference Noise Levels 56 Figure 3.8 Minimum Magnitude of the Edges for Identification under Difference Noise Levels . 57 Figure 3.9 Magnitude of the Edges Identification with 100% Confidence at Difference Noise Levels . 58 Figure 3.10 Piecewise Constant signal and the Edge Points at (71,199,243,420,469) . 60 Figure 3.11 Likelihood Function of the Signal Given Difference Model Order 60 Figure 3.12 Confidence of Increasing Model Order and the Threshold of 75% . 60 Figure 3.13 Posterior Evidence of the Given Signal with Corresponding Identified Edges at (71,199,243,420,469) . 61 Figure 3.14 2-D Image for the Edge Detect 62 Figure 3.15 Comparison of Edge Detect by Proposed Method and Sobel Detector 63 ME Department of National University of Singapore VII Weimiao YU Bayesian Approaches for Image Restoration Figure 3.16 Restored Images by Proposed Method and Adaptive Wiener Filter 63 Figure 4.1 Distribution of α with Different Parameters 69 Figure 4.2 Roof Edge Experimental Result: Noise ! = 10. Total Discontinuous Points in the Signal; Number of Used Kernels is 9; Estimated Noise Level !ˆ = 9.63. RMS of the Fitting is 2.64; L!f ' =6.97. Average of LR at Those Points is 10% 76 Figure 4.3 Roof Edge Experimental Result: Noise ! =40. Total Discontinuous Points in the Signal; Number of Used Kernel is 7, Estimated Noise Level is !ˆ = 39.10. RMS of the Fitting is 39.26; L!f ' =23.32. Average of LR at Those Points is 17% . 76 Figure 4.4 Step Edge Experimental Result: Noise ! =10. Total Discontinuous Points in the Signal; Number of Used Kernel is 14; Estimated Noise Level !ˆ = 9.67. RMS of the Fitting is 2.63; L!f ' = 12.14. Average of LS at Those Points is 3% 78 Figure 4.5 Step Edge Experimental Result: Noise ! =40; Total Discontinuous Points in the Signal; Number of Used Kernel is 9; Estimated Noise Level is !ˆ = 39.19. RMS of the Fitting is 13.84; L!f ' = 87.069. Average of LS at Those Points is 22.5% 78 Figure 4.6 Two Types of Edges Experimental Result: Noise ! =10. Total Step Edge and Roof Edges in the Signal; Number of Used Kernel is 13; Estimated Noise Level !ˆ = 10.67. RMS of the Fitting is 5.03; L!f ' = 32.78. Average of LS and LR at Those points is 16% . 80 Figure 4.7 Two Types of Edges Experimental Result: Noise ! =40. Total step Edge and Roof Edges in the Signal. Number of Used Kernel is 8; Estimated Noise Level !ˆ = 39.53. RMS of the Fitting is 11.12; L!f ' = 73.68. Average of LS and LR at Those Points is 13% 80 Figure 5.1 Local Piecewise Constant and Piecewise Linear Assumptions for the Cameraman. (a). The Rectangle Indicates the Window Size. (b) Shows the Red Rectangle Portion in 3D. (c) Shows the Blue Rectangle Portion in 3D. (Window Size=10) 85 Figure 5.2. Local Piecewise Constant and Piecewise Linear Assumptions for the Lena. (a). The Rectangle Indicates the Window Size. (b) Shows the Red Rectangle Portion in 3D. (c) Shows the Blue Rectangle Portion in 3D. (Window Size=10) . 86 Figure 5.3 The Kernel Matrix Applied in Learning Process (η=1.0) . 89 ME Department of National University of Singapore VIII Weimiao YU Bayesian Approaches for Image Restoration Cartesian versions are displayed in Figure 7.5. The size of the images is 256×256 and the center of rotation is at the centre of the image, or at (128, 128). In this case, the optimized ! r and "! are 0.45 and 0.33° respectively, according to the steepest gradient optimization of the PRR. The non-blurred images in Cartesian coordinates are shown on the top and the simulated blurred images are at the bottom. The PSF in Pz is assumed to be a constant angular velocity in the simulations. For more details about the more complicated PSF, which is not a constant angular velocity, please refer to [1]. All experimental results displayed in Figure 7.5 use the optimized PRR according to Eq. (7.17). When non-optimized PRR is applied for the transformation, there will be some artificial artifacts. The simulated blurred image will be discontinuous at some points in the Cartesian coordinates if the transformation is applied without optimizing the PRR. It is another interesting topic to discuss blurred image using the transformation with and without optimization of the PRR. However, this is beyond the scope of the current work. In the next example, the noisy rotational blurred image of “Camera Man” is simulated and then restored in Polar coordinate domain. Finally, the restored image is shown in the Cartesian coordinates by using the inverse Polar transform. This procedure is also shown in Figure 7.6. The original image of the “Camera Man” is shown in Figure 7.6(a). The SPs of the image are shown in Figure 7.6(b). After the interpolation the image in Polar coordinate domain is shown in Figure 7.6(c). And its noisy rotational blurred version in the same domain is shown in Figure 7.6(d). The noisy rotational blurred image in Cartesian coordinates is shown in Figure 7.7(a). The denoised image in Polar coordinate is shown in Figure 7.7(b). The deblurred image in Polar coordinate domain is shown in Figure 7.7(c); and finally, the restored image in Cartesian coordinate domain is shown in Figure 7.7(d). ME Department of National University of Singapore 150 Weimiao YU Bayesian Approaches for Image Restoration (a). Original Image of Cameraman (b). SPs in Discrete Polar Domain (c). Image in Discrete Polar Domain (d). Blurred Image in Polar Domain Figure 7.6. Original Image of Cameraman, the SPs in Polar Domain, the Image in Polar Domain and the Blurred image in Polar Domain (Blurred Degree=18°) (a). Blurred Image in Cartesian Domain (b) Denoised Image in Polar Domain (c). Restored Image in Polar Domain (d) Restored Image in Cartesian Domain Figure 7.7 The Restoration of Rotational Blurred Image in Polar Domain and the Restored Image in Cartesian Domain ME Department of National University of Singapore 151 Weimiao YU Bayesian Approaches for Image Restoration From this simulation, the Compact Discrete Polar Coordinates transform can successfully change a spatially variance problem of rotational blurred image to a more manageable spatially invariant problem in the new Polar domain. Both the blurring and deblurring procedures could be successfully simulated in Polar coordinates. The inverse transformation can also convert the image in Polar coordinate back to the Cartesian coordinates and achieve good visual results. Figure 7.8 Setup of the Rotational Motion Platform and Imaging System In order to test presented approach in real rotation blurred image restoration, an experimental platform was established by Mr. Lee Shay Liang[116], as shown in Figure 7.8. The calibration of the camera was done by Mr. Ng Wei Hiong [117]. (a) Image for the Rotation Centre Calibration (b) Blurred Image for Calibration Figure 7.9 Calibration of Rotation Centre ME Department of National University of Singapore 152 Weimiao YU Bayesian Approaches for Image Restoration Since the rotational centre should be known to apply the coordinates transformation, the rotational centre is obtained by the camera calibration instead of using other rotational centre identification methods to avoid the influence of the error of the location of the rotational centre. The centre of rotation is easily obtained by the artificial image. The image for calibration and the blurred image are shown in figure 7.9. After the calibration, the rotation center is found to be at (126, 127) in the Cartesian coordinates. The image is mounted on the platform and then captured by a camera, which is in rotational motion. The captured images are shown in Figure 7.10(a) and 7.10(c). (a) Rotational Blur Image of “Star” (c) Rotational Blur Image of “Cameraman” (b) Restored Image by Proposed Approach (b) Restored Image by Proposed Approach Figure 7.10 Restoration of the Real Rotation Blurred Images ME Department of National University of Singapore 153 Weimiao YU Bayesian Approaches for Image Restoration From the above simulations and experiments, the Compact Discrete Polar transformation is a good approach to simplify the spatially variant problem to a spatially invariant one. It can successfully restore the rotational blurred image. In the new Compact Polar coordinate domain, the spatially invariant formulation could be ready to solve the rotational blurred image, which is spatially variant in Cartesian coordinates. 7.6. Summary In this chapter, a Discrete Compact Polar Coordinates Transformation is developed to handle the image restoration task of a rotational blurred image. By defining the coordinates transformation carefully, the initial spatially variant problem could be converted and simplified to a spatially invariant problem in the new coordinates. The transformation in continuous domain is straightforward; however, additional issues in the discrete domain are considerable. The linear interpolation is applied to deduce the values of the VPs for the consideration of computational efficiency based on the assumption made during sampling. The constraint of Sufficiency of the transformation is relaxed and replaced by a weaker constraint, i.e. the optimization of the PRR and minimization of the number of VPs. Since the number of the “empty points” is minimized, the transformation is so-called Compact Discrete Polar Transformation. From the experimental results, images in discrete Cartesian lattices can be successfully rotationally blurred by simulation based the proposed transformation. The invariance and variance of the rotational blur problem are separable in the compact polar coordinates transform. Two real rotational blurred images captured by the established platform are used to validate the proposed approach. Both the simulation and the experiments achieved excellent results. The proposed coordinates transformation has two advantages, uniform sampling and compact format. Uniform sampling allows the traditional spatially invariant image identification and ME Department of National University of Singapore 154 Weimiao YU Bayesian Approaches for Image Restoration restoration algorithms in Cartesian coordinates to be readily applied to the spatially variant problem in the new coordinates by variable replacement according to the three steps stated in section in this Chapter. The compactness will reduce the under-sampling in the Polar coordinates. The linear interpolation method is applied to deduce the gray level of the virtual points based on the assumptions made on the sampling interval in Cartesian. ME Department of National University of Singapore 155 Weimiao YU Bayesian Approaches for Image Restoration 8. Chapter Conclusion The objective of this dissertation is to develop approaches to handle image restoration problem for images blurred by motion of the camera or the objects in questions. Bayesian approaches have been chosen due to its robustness and accuracy. Its solid and established statistical foundation based on which efficient theoretical approaches could be built to solve our problems at hand. Necessary mathematical representations of images have been established in Chapter 2. They are consistently used in the subsequent development of the proposed approaches and algorithms. Due to the importance of discontinuities in providing useful cue for the image restoration work, this work began with the identification of the edge points using the proposed approach on 1-D piecewise constant signals in order to gather better understanding of the potential difficulties at hand. Posterior evidence is proposed to select model orders since the prior distribution function is not proper. By assuming that the model is a Binomial distribution, the calculation of the evidence is simplified in Bayesian theory, and to avoid evaluating the posterior marginal likelihood function directly. Simulation and experiments proved that this approach is robust and accurate. Subsequently, a novel Bayesian kernel approach on 1-D and 2-D discontinuous signal/image restoration is presented. Based on the assumption that the signals are Local Piecewise Linear, noise was removed and the discontinuities were identified and preserved. a novel convex cost function for image noise removal therefore is proposed. This doing helped to resolve the illconditioned problem of image restoration. The proposed cost function is proven to have only one global minimum. The cost function was minimized by tuning a proper hyper-parameter η through the steepest gradient optimization algorithm. The results are compared with those of LPA-ICI approach, as the latter is known to be recent and good. There are some other existing methods, ME Department of National University of Singapore 156 Weimiao YU Bayesian Approaches for Image Restoration but their performance is not comparable to the proposed method nor that of LPA-ICI. The proposed approach achieved similar results: ISNR and PSNR is slightly low than ICI, the 1- and 2- Norm error are close, and the Infinite norm error is much smaller than that of LPA-ICI when noise level is high, i.e. at low SNR. However, it is slower than LPA-ICI, this is because of the necessity in evaluating the inverse of the Hessian matrix. LPA-ICI, on the other hand, requires convolution and de-convolution computation, which could be fast by using the well established FFT in the calculation. Restoration of images blurred by motion of the camera and/or objects in the scene is also one of the aims of this dissertation. After the noise has been successfully removed with the edges preserved, the characteristics of the Point Spread Function (PSF) should be gathered with ease. The PSF will help to identify the motion blur, based on which the degraded image could be restored. A statistical approach for motion blur identification was presented in Chapter 6. Experimental results demonstrated its effectiveness even under low SNR conditions. It is also robust and accurate. Finally, a more difficult problem is studied, the restoration of images under spatially variant condition. This would be the case when an image is blurred by rotation motion. In this situation, a normal approach would be to represent every pixel by a PSF before any restoration work could be performed. This is obviously not feasible as the computational work would be fastidious. With the innovative approach using a novel coordinate transformation, the spatially variant problem is reduced to a spatially invariant one. The formulation of the Discrete Polar Coordinates Transformation and its inverse are presented in Chapter 7, and it has two advantages, viz. Uniform Sampling Interval and Compactness (less missing data points after coordinate transformation). The Uniform Sampling allows the existing approaches of blur identification and image restoration methods to be applied to even rotational blur problem. Note that blurring due to rotational blur is originally a spatially variant problem. Experimental results convincingly show ME Department of National University of Singapore 157 Weimiao YU Bayesian Approaches for Image Restoration the effectiveness of the proposed method. Furthermore, simulation shows that Compact Discrete Polar Transformation also can be applied for the problem of out-of-focus blurring problem, which is 1-D spatially invariant in the Polar domain. However, results are not presented in this dissertation due to the limitation of space. For future works, some recommendations are as following: 1. The use of “traveling windows” in Chapter for noise removal requires further refinement. Further works should be carried out to determine the size of the window in relation to the quality of the image in question. A correct size of the window will help to speed up the computation and hence the procedure as a whole. In addition, a global GML representation of image, without using windows, is also desired, but, an efficient algorithm with a manageable size of data must be developed. 2. A more accurate global noise estimation method and the associated computation algorithm are needed. With more accurate noise variance estimation, further performance improvement of the proposed restoration approach can be achieved. This will help in the optimization of the hyper-parameter η. Furthermore, new or improved optimization technique should be developed to reduce the residue of Cost function that theoretically should be zero. 3. In the work on the restoration of images blurred by rotational motion, the centre of rotation was based on calibration technique, which might not be accurate in some situations. A proper calibration technique should be established to estimate accurately the centre of rotation. 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ME Department of National University of Singapore 165 [...]... previous works are also included in the individual chapters for their relevance and for easy comparisons ME Department of National University of Singapore 8 Weimiao YU Bayesian Approaches for Image Restoration 1.3.1 Image Identification Image identification is an indispensable step in image restoration It provides necessary information for image restoration Typically, the PSF and the noise variance are... 5.24 RMS Error of Bayesian Kernel, LPA-ICI and Wiener Filter (Image of “Lamp”) 110 Figure 5.25 ISNR of Bayesian Kernel, LPA-ICI and Wiener Filter (Image of “Egg”) 110 Figure 5.26 MAX Error of Bayesian Kernel, LPA-ICI and Wiener Filter (Image of “Lena”) 110 ME Department of National University of Singapore IX Weimiao YU Bayesian Approaches for Image Restoration Figure... popular and active methods for image restoration TV based regularization is a typical case of geometry-driven diffusion for image restoration A reliable and efficient computational algorithm for image restoration is proposed in [65] A piecewise linear function (a measure of total variation) is minimized subject to a 2-norm inequality constraint (a measure of data fit) for discrete image The blur is removed... Continuous Coordinates Transform and Its Inverse !1 TZ / TZ : Discrete Coordinates Transform and Its Inverse ! r / "! : Polar Resampling Resolution for r and θ; S n (r | ! r ) / S n (! | "! ) : Sampling Operation in Polar Domain for Given ! r and "! ; Г(l, k): Image in Discrete Polar Domain; ME Department of National University of Singapore XV Weimiao YU Bayesian Approaches for Image Restoration LIST OF ABBREVIATION... since nearly all the images are degraded by noise Generally, image restoration is defined as the problem of recovering an image from its blurred or/and noisy rendition for the purpose of improving its quality Therefore, three kinds of image restoration problems are commonly encountered: (a) Restore image from its blurred version when Signal to Noise Ratio (SNR) is high; (b) Restore image from it noisy... restore images blurred due to rotational motion of the camera will be developed The procedures are summarized in Figure 1.1 Figure 1.1 The Procedure of Image Restoration The main steps after a blurred image has been captured are: 1 Noise removal 2 Checking whether blurred image is caused by rotational motion? ME Department of National University of Singapore 5 Weimiao YU Bayesian Approaches for Image Restoration. .. processing is done in the new transformed coordinates The concept of compactness will be proposed A significant advantage of the proposed novel approach in coordinates transformation is that the image is still uniformly sampled in the new polar coordinates, therefore, the existing image identification and restoration approaches can be easily implemented Blur Identification and Image Restoration Blur identification... 151 Figure 7.7 The Restoration of Rotational Blurred Image in Polar Domain and the Restored Image in Cartesian Domain 151 ME Department of National University of Singapore X Weimiao YU Bayesian Approaches for Image Restoration Figure 7.8 Setup of the Rotational Motion Platform and Imaging System 152 Figure 7.9 Calibration of Rotation Centre 152 Figure 7.10 Restoration of the... Weimiao YU Bayesian Approaches for Image Restoration LIST OF SYMBOLS f(x, y)/ f(i, j): Original Continuous Image/ Digital Image; g(x, y)/g(i, j): Observed Continuous/Digital Blurred Image; d(x, α, y, β): General PSF; en(x, y)/en (i, j): Environment Noise; em (x, y)/em (i, j): Modeling Noise; Q: Covariance of Environment Noise; !: Support of the Image; !d : Support of the PSF;SP G(v, u): Fourier Transform... of Blurred Image from p l y (SNR=40, T for Different Images is Indicated in the Corresponding Tables PSF is the Uniform Linear Motion Blur p l y s of the Original Images are Monotonically Decreasing, as Shown in Figure 6.3.) 127 Figure 6.9 PSF Extent Identification Results of Blurred Image from p l y (SNR=40, T for Different Images is Indicated in the Corresponding Tables PSF is the Uniform Linear . blurred image restoration. Weimiao YU Bayesian Approaches for Image Restoration ME Department of National University of Singapore VII LIST OF FIGURES Figure 1.1 The Procedure of Image Restoration. the Weimiao YU Bayesian Approaches for Image Restoration ME Department of National University of Singapore VI shape of the PSF, so it can be applied for any shape of PSF. The image is decomposed. 3.3.2 Experiments for the Single Edge Point Model 54 Weimiao YU Bayesian Approaches for Image Restoration ME Department of National University of Singapore III 3.3.3. Experiments for the Multi